Best Known (51, 51+8, s)-Nets in Base 4
(51, 51+8, 65546)-Net over F4 — Constructive and digital
Digital (51, 59, 65546)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 10)-net over F4, using
- digital (46, 54, 65536)-net over F4, using
- net defined by OOA [i] based on linear OOA(454, 65536, F4, 8, 8) (dual of [(65536, 8), 524234, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(454, 262144, F4, 8) (dual of [262144, 262090, 9]-code), using
- 1 times truncation [i] based on linear OA(455, 262145, F4, 9) (dual of [262145, 262090, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 418−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(455, 262145, F4, 9) (dual of [262145, 262090, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(454, 262144, F4, 8) (dual of [262144, 262090, 9]-code), using
- net defined by OOA [i] based on linear OOA(454, 65536, F4, 8, 8) (dual of [(65536, 8), 524234, 9]-NRT-code), using
(51, 51+8, 262175)-Net over F4 — Digital
Digital (51, 59, 262175)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(459, 262175, F4, 8) (dual of [262175, 262116, 9]-code), using
- 1 times truncation [i] based on linear OA(460, 262176, F4, 9) (dual of [262176, 262116, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- linear OA(455, 262144, F4, 9) (dual of [262144, 262089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(45, 32, F4, 3) (dual of [32, 27, 4]-code or 32-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(8) ⊂ Ce(4) [i] based on
- 1 times truncation [i] based on linear OA(460, 262176, F4, 9) (dual of [262176, 262116, 10]-code), using
(51, 51+8, large)-Net in Base 4 — Upper bound on s
There is no (51, 59, large)-net in base 4, because
- 6 times m-reduction [i] would yield (51, 53, large)-net in base 4, but